Full waveform inversion and the inverse Hessian

Gary F. Margrave, Matthew J. Yedlin, Kristopher A. H. Innanen

Full waveform inversion involves defining an objective function, and then moving in steps from some starting point to the minimum of that objective function. Gradient based steps have long been shown to involve seismic migrations, particularly, migrations which make us of a correlation-based imaging condition. More sophisticated steps, like Gauss-Newton and quasi-Newton, alter the step by involving the inverse Hessian or approximations thereof. Our interest is in the geophysical, and practical, influence of the Hessian. We derive a wave physics interpretation of the Hessian, use it to flesh out a published statement of Virieux, namely that performing a quasi-Newton step amounts to applying a gain correction for amplitude losses in wave propagation, and finally show that in doing so the quasi-Newton step is equivalent to migration with a deconvolution imaging condition rather than a correlation imaging condition.